The use of shaft systems to transmit torque from input sources (e.g. engines) to output devices (e.g. gearboxes) is well known in the art. Typically, design parameters of a shaft system require that the shaft system operates within a rotational frequency range known as an operating frequency range, defined by a lower operating frequency and an upper operating frequency.
When a shaft system operates at high rotational frequencies, the shaft system passes through progressively higher rotational frequency ranges known as critical frequencies. At these critical frequencies, the physical properties of the shaft system and the rotational frequency of the shaft are such that a predominant vibration response occurs at a resonance frequency of the system. The resonance frequency is defined as a frequency at which the vibration response is at a maximum. The critical frequencies are referred to in order of their appearance, i.e., first order critical frequency, second order critical frequency.
Although many shaft systems contain features that may distinguish their critical frequencies from that of other shaft systems, the following general equation may be used to ascertain a critical frequency for a shaft that is simply supported at each end and has a constant cross-sectional area: ##EQU1## wherein; C=constant (1.57 for the first order critical frequency, 6.28 for the second order critical frequency)
g=gravitational acceleration; PA1 E=modulus of elasticity; PA1 I=cross-sectional moment of inertia; PA1 w=weight per unit length of the shaft; and PA1 L=length of the shaft. PA1 defining an adjusted first order critical frequency for the rotating shaft, the adjusted first order critical frequency being equal to or less than the lower operating frequency; and PA1 disposing a mass having a predetermined weight in combination with the rotating shaft at the second order critical frequency node, whereby the mass maintains the adjusted first order critical frequency equal to or less than the lower operating frequency, thereby controlling lateral vibration of the rotating shaft as the rotating shaft rotates within the operating frequency range.
When a shaft system operates at or near a critical frequency, the shaft assumes a deflected shape known as a mode shape. The mode shape reflects lateral deflections of the shaft relative to the shaft's axis of rotation. A node in the mode shape is a location at which no deflections occur.
FIG. 1 depicts a typical shaft system comprising a shaft S supported at both ends thereof by bearings R, driven by an input I and connected to an output O. As depicted in FIG. 1, since the shaft S is not supported intermediate of its ends, the first order critical frequency of the shaft S is characterized by the shaft S assuming a sinusoidal mode shape approximating a half sine wave. When the shaft S assumes this mode shape, the shaft rotation becomes unstable and excessive vibration may occur. This vibration may damage elements of the shaft system, including the bearings, bearing support structures, and the shaft itself. When the rotation of the shaft S is increased beyond the first order critical frequency, the vibration subsides and the shaft S assumes a smoother rotation.
The shaft S will exhibit this smooth rotation until the frequency of the shaft reaches the second order critical frequency. As depicted in FIG. 2, the second order critical frequency is characterized by the shaft S assuming a sinusoidal mode shape approximating a full sine wave. When the shaft S assumes this mode shape, a second order critical frequency node N is defined substantially near the longitudinal center of the shaft S. When the shaft S assumes the second order critical frequency mode shape, the shaft rotation once again becomes unstable and may result in excessive vibration.
In a typical shaft system, the second order critical frequency occurs at a rotational frequency higher than the upper operating frequency such that the operation of the shaft system is not affected. By contrast, it is common in many shaft systems for the first order critical frequency to occur within the operating frequency range, thereby introducing the potential for shaft instability and the undesirable vibratory characteristics characterized above. Since the location of the first order critical frequency within the operating frequency range is undesirable, the prior art discloses a number of techniques for raising the first order critical frequency to a point beyond the operating frequency range.
Referring to the equation above for defining a critical frequency of a shaft system, it will be appreciated that since the cross-sectional moment of interia (I) is a function of the diameter of the shaft, it follows that the first order critical frequency is proportional to the diameter of the shaft. Therefore, if the diameter of a shaft is increased, the first order critical frequency will proportionally increase as well. However, increasing the diameter of the shaft also increases the weight of the shaft, and may make the shaft more expensive to manufacture. These weight and cost increases may not be acceptable, nor feasible, depending on the application for the shaft system.
It will also be appreciated that the first order critical frequency is inversely proportional to the square of the length of the shaft. Therefore, for example, decreasing the shaft length by 1/2 results in a 4.times. increase in the first order critical frequency. Two primary methods have been used in the prior art to decrease the length of a shaft.
The first method comprises using coupling or bearing assemblies to linearly link a plurality of short shafts together, in series, along a common axis of rotation. The disadvantages of this method are that it adds weight and expense to the shaft system, and may require increased maintenance to maintain the couplings and bearings in good working order.
The second method comprises using a long shaft with multiple bearing assemblies disposed along the length of the shaft such that the bearings prevent lateral displacement of the shaft at those points. The effect of this method is that the shaft behaves as if it was a plurality of shorter shafts linked linearly together. The disadvantages of this method are similar to those above, in that the plurality of bearings adds weight and cost to the assembly, and may necessitate increased maintenance to maintain the bearings in good working order.
U.S. Pat. No. 4,217,766 to Suckow discloses another method for increasing the first order critical frequency. In Suckow, an axial collar is placed substantially coaxial with a shaft and is fixed with respect thereto for rubbing contact with the shaft when the shaft deflects a predetermined amount from its rotational axis. The rubbing contact alters the bending mode of the shaft, thereby increasing the first order critical frequency to a level above a maximum frequency for the shaft. A disadvantage of this method is that the rubbing between the shaft and the collar necessitates deflection of the shaft as a condition precedent for vibration control within the operating frequency range.